The anatomy, location, and the flexibility of the shoulder joint makes it a very tough spot to compress.
I have had two shoulder surgeries and have made successful recovery on both counts. I did rock a compression sleeve thingy working my way back from surgery/rehab. And honestly, it did next to nothing. I especially hated the wrap-around strap and how it irritate my off-armpit.
Icing works well enough for me. Nothing fancy, slip an ice-pack underneath a jacket or sweater after playing. And that's just to keep the ice-pack down.
$5-8 ice-pack is good enough.